English

M S Mahdy A,S Mohamed A,A H Mtawa A

doi:10.5897/ijps2015.4287

Abstract

We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. This method is called the variational homotopy perturbation method. We use this method for solving Generalized Time-space Fractional Schrödinger equation. The fractional derivative is described in Caputo sense. The proposed scheme finds the solution without any discritization, transformation or restrictive assumptions. Several example is given to check the reliability and efficiency of the proposed technique.   Key words: Caputo derivative, variational iteration method, homotopy perturbation method, Schrödinger equation.

Highlights

We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method
We investigate the application of the VARIATIONAL HOMOTOPY PERTURBATION METHOD (VHPM) for solving the generalized time-space fractional Schrödinger equation with variable coefficients (Rida et al, 2008; Ganjiani, 2010): i
If we select 1, v(x) 0, this equation turns to the famous nonlinear Schrödinger equation in optical fiber (Hao et al, 2004; Chen and Li, 2008; Li and Chen, 2004)

Summary

Introduction

Considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of nonlinear science (Dalir and Bashour, 2010), many important phenomena (Podlubny, 1999), engineering and physics (Miller and Ross, 1993), dielectric polarization (Sun et al, 1984), quantitative finance (Laskin, 2000).To find explicit solutions of linear and nonlinear fractional differential equations, many powerful methods have been used such as the homotopy perturbation method (Momani and Odibat, 2007; Wang, 2008; Gupta and Singh, 2011), the Adomain decomposition method (Ray, 2009; Herzallah and Gepreel, 2012; Rida et al, 2008), the variational iteration method (He, 2000, 2004, 2007; He and Wang, 2007), the homotopy analysis method (Hemida et al, 2012; Gepreel and Mohamed, 2013; Ganjiani, 2010; Behzadi, 2011), the fractional complex transform (Ghazanfari, 2012; Su et al, 2013), the homotopy perturbation Sumudu transform method (Karbalaie et al, 2014; Mahdy et al, 2015), the local fractional variation iteration method (Yang and Baleanu, 2013; He and Liu, 2013; Yang et al, 2014), the local fractional Adomain decomposition method (Yang et al, 2013b), the Cantor-type Cylindrical-Coordinate method (Yang et al, 2013c), the variational iteration method with Yang-Laplace (Liu et al, 2013), the Yang-Fourier transform (Yang et al, 2013a), the Yang-Laplace transform (Zhao et al, 2014; Zhang et al, 2014) and variational homotopy perturbation method by (Noor and Mohyud-Din,2008). We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. We use this method for solving Generalized Time-space Fractional Schrödinger equation. We investigate the application of the VHPM for solving the generalized time-space fractional Schrödinger equation with variable coefficients (Rida et al, 2008; Ganjiani, 2010): i

Results

Conclusion